Part 1
Because he split his soul too often, Voldemort actually died when Lily's protection threw his curse back at him. Harry is not a Horcrux, but the other ones are still around, but acting comparable to the diary or how the medallion affected the trio rather than respawn beacons.
Year one happened as it did, but Quirrel is acting for some underground group who want infinite gold. Year two happens as it does in canon, as does year 3. Year 4 is canon, but Harry is entered to kill him in a humiliating way rather than for the graveyard plot. He gets no help and struggles more. Year 5 has Umbridge in Hogwarts due to the Fudge/Dumbledore powerstruggle and the falling standards. It ends with Umbridge exposed and Fudge thrown out due to the scandal.
Here's where things get interesting. Lucius Malfoy is elected as new Minister and starts to make subtle changes. Certain wordings of laws which can be used by the wizengamot and such. He does not ban Muggleborn from high positions but creates a system which looks fair at first glance, with the devil in the detail. Most of books 6 and 7 pass that way.
Part 2
Now enter Harry, who has just passed his NEWTS and turned 18, taking up a seat. He is promptly thrown under the bus by more experienced politicians and left looking a fool in debates since he never had proper oratory training. So he makes Hermione his aide, which helps him but takes a lot of effort. Andromeda could also fill this role, or be an additional advisor. One evening, he takes a break and goes to the Leaky for a pint or five, where he runs into Fleur, the persona non grata of the Weasley family after she broke off the engagement after the Weasleys were their canon selves. They have a few drinks together and agree to meet again next week.
Harry lowkey panics since he does not know whether that meeting will be a date or not. They meet, have fun and agree on a repeat, again not defining if date or not.
Basically, a romance story while Harry has to learn the political stuff fast enough to stop Malfoy's slow burn takeover of the magical world and do something against the people who supported and enabled Malfoy. Maybe have methods get dirtier down the road, with mafia style extortion, intimidation and such, where Malfoy uses some of the muscle of the old gang for that while the better Death Eaters are helping him in the Wizengamot.
Part 3
Because they struggle economically, Muggleborns turn to crime and build their own district in one of the poorer areas of Diagon Alley. They Band together to protect their shops, offer loans, help people to find jobs and such. Malfoy sends his thugs against them, sparking a gang war in the poorer parts of magical London. Some people smell a profit and open shops in "hostile" territory, content to pay protection money, others pretend to do so and spy.
To counter Malfoy's takeover of the underground, Harry has to work together with them, maybe even usurping one of Muggleborn gangs. Tonks would be ideal for the kind of infiltration/spy arcs which would follow. Fleur on the other hand would have connections to France or the Veelas which would be useful, while publicly playing Harry's better side. To counter French and or Veela involvement, Lucius uses his own underground connections to Ireland and/or Scandinavia.
In the end, Harry might have to turn on one of the Muggleborn gangs because they crossed a line (too much collateral damage for example, or selling hard drugs).
For the setting, this could include the run down parts of muggle London as well as a darker version of Diagon. Abandoned railway and underground tunnels, closed industrial estates, old warehouses and so on.